Inverse Operator Research Articles

Coefficients of Exponential Series for Analytic Functions and the Pommiez Operator

In this paper, we present results of the existence of a linear continuous right inverse operator for the operator of the representation of analytic functions in a bounded convex domain of the complex plane by series of quasi-polynomials and exponentials. We also present closely related results on the A. F. Leontiev interpolating function and, more generally, on the interpolating functional and the corresponding Pommiez operator. We examine cyclic vectors and closed invariant subspaces of the Pommiez operator in weighted spaces of entire functions.

On the compactness of the resolvent of a Schrödinger type singular operator with a negative parameter

In this paper, we consider a Schrödinger-type operator with a negative parameter and a complex potential Lt=−Δ+(−t2+itb(x)+q(x)),x∈Rn,n≥1,i2=−1, where t is a parameter that arises when studying hyperbolic operators in the space L2(Rn+1). We assume with respect to the coefficients of the operator Lt that they are continuous in Rn strongly growing and rapidly oscillating functions at infinity and satisfy the condition |b(x)|≥δ0>0,q(x)≥δ>0.In the paper, under these assumptions, it is proved that there exists a bounded inverse operator for all t∈R and found a condition that ensures the compactness of the resolvent.Note that in the paper we construct regularizing operator, i.e. the question of the existence of a resolvent of an operator with singular coefficients reduces to the case of operators with smooth periodic coefficients.

Proper inverse operators of fractional derivatives with nonsingular kernels

Proper inverse operators of fractional derivatives with nonsingular kernels

The Laguerre transform of a convolution product of vector-valued functions.

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions.
 The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences.
 The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

Approximation by interpolation spectral subspaces of operators with discrete spectrum

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions.
 The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences.
 The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

Trefftz Method of Solving a 1D Coupled Thermoelasticity Problem for One- and Two-Layered Media

This paper discusses a 1D one-dimensional mathematical model for the thermoelasticity problem in a two-layer plate. Basic equations in dimensionless form contain both temperature and displacement. General solutions of homogeneous equations (displacement and temperature equations) are assumed to be a linear combination of Trefftz functions. Particular solutions of these equations are then expressed with appropriately constructed sums of derivatives of general solutions. Next, the inverse operators to those appearing in homogeneous equations are defined and applied to the right-hand sides of inhomogeneous equations. Thus, two systems of functions are obtained, satisfying strictly a fully coupled system of equations. To determine the unknown coefficients of these linear combinations, a functional is constructed that describes the error of meeting the initial and boundary conditions by approximate solutions. The minimization of the functional leads to an approximate solution to the problem under consideration. The solutions for one layer and for a two-layer plate are graphically presented and analyzed, illustrating the possible application of the method. Our results show that increasing the number of Trefftz functions leads to the reduction of differences between successive approximations.

Existence of positive solutions for Brezis-Nirenberg type problems involving an inverse operator

This article concerns the existence of positive solutions for thesecond order equation involving a nonlocal term $$ -\Delta u=\gamma (-\Delta)^{-1} u+|u|^{p-1}u, $$ under Dirichlet boundary conditions. We prove the existence of positive solutions depending on the positive real parameter \(\gamma>0\), and up to the critical value of the exponent \(p\), i.e. when \(1<p\leq 2^*-1\), where \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent. For \(p=2^*-1\), this leads us to a Brezis-Nirenberg type problem, cf. \cite{BN}, but, in our particular case, the linear term is a nonlocal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant, that ensures the existence of positive solution, going from dimensions \(N\geq 4\) in the classical Brezis-Nirenberg problem, to dimensions \(N\geq7\) for this nonlocal problem. For more information see https://ejde.math.txstate.edu/Volumes/2021/52/abstr.html

A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems

Abstract We propose a class of Newton-like methods with increasing convergence order for approximating the solutions of systems of nonlinear equations. Novelty of the methods is that in each step the order of convergence is increased by an amount of three at the cost of only one additional function evaluation. Another important feature is the single use of an inverse operator in each iteration, which makes the schemes attractive and computationally more efficient. Theoretical results regarding convergence and computational efficiency are verified through numerical examples including those arising from boundary value problems and integral equations. By way of comparison, it is shown that the present methods are more efficient than their existing counterparts, particularly when applied to solve the large systems of equations.

WITHDRAWN: Study of a computational model for a differential equation with empirical functions based on the integral equations Fredholm of the first kind

The work out lines a method of constructing an approximate solution of a differential equation of the second order with input data obtained in the experiment (empirical functions). In such statement the problem belongs to the class of incorrect mathematical problems and often occurs, for example, in mathematical models of physical phenomena using measurement results of field experiments. To obtain the approximate solution of this problem requires construction of appropriate regularization algorithms based on the methods of the theory of functional analysis and ill-posed problems. In the present work is the construction of the approximate solution of odes with specified boundary conditions, are the so-called singular integrals. This allows you to put in the original equation Fredholm integral equation of the first kind and to find its numerical solution, i.e. the solution of the incorrect task. This uses a machine approximation of functions and their deriva- tives corresponding singular integrals and regularization method convergence of the sequence of approximate solutions, which implemented the so-called generalized inverse operators. Built in the end, a computational model allows to obtain a stable solution of ill-posed problems.

An operator approach for modelling the interdependence between the soil water boundary wetting and drying curves

In this study the relationship between two families of pairwise associated functions, describing boundary wetting and drying curves, is modelled. This relationship includes both regular and random constituents, and it is represented by a limited dataset of N known pairs of these curves, which pertain to N different soils. Using the dependent domain theory of hysteresis, we show that the regular constituent in this relationship can be described by a linear integral operator that, for a given input measured boundary wetting curve, predicts its associated boundary drying curve. The operator is optimized to reduce the influence of the random scattering of the pairs of curves. The optimization of the operator is performed with respect to the given input measured boundary wetting curve using a representative calibration dataset of k (k≤N) pairs of curves, which are a part of the N available measured pairs. The optimized operator should satisfy two opposite conditions: (i) the best possible accuracy at reproducing the boundary drying curves used for operator optimization, and (ii) the least possible sensitivity with respect to excluding any one pair from the calibration dataset.We also apply the suggested modelling approach for derivation of the inverse operator model that predicts the boundary wetting curves from their associated measured boundary drying curves, with possibility of verification of the predictive reliability. The predictive performance of both operator models is mainly good, and their reliability can be permanently enhanced by updating them with new incoming measured data.

HDR-Like Image from Pseudo-Exposure Image Fusion: A Genetic Algorithm Approach

Image distortion during multiple-exposure capture is a common scenario, and most of the existing legacy image content is in the low dynamic range (LDR). For both of these problems, pseudo multiple exposure image generation from a single image is a possible solution. Hence, in this article, we propose a novel method for high dynamic range (HDR)-like image generation by fusing pseudo-multi-exposure images that are produced from only one single image. The proposed pseudo-exposure image generation method is based on intensity mapping function (IMF). First, our system learns the relation between different exposure images with the help of genetic algorithm and generates long and short exposure images. In the second step, we fuse images with different exposures to generate the HDR-like image. Our method outperforms several inverse tone mapping operator (ITMO)-based methods. The simplicity and good performance of the proposed method makes it suitable for mobile phones and other consumer electronics. We also develop a demo software to validate the proposed method.

Migration Deconvolution via Deep Learning

Migration deconvolution is an image domain approach to least-squares migration, which is considered the state-of-the-art algorithm for obtaining seismic reflectivity models of the earth from seismic acquisition results. Seismic imaging is an active research field with the development over the last few years of several techniques that have mitigated imaging issues. Ongoing research aims to improve image resolution and thus provide a more reliable seismic amplitude for the interpreter. Migration deconvolution can be framed as an inverse problem in the image domain to mitigate image resolution problems and reduce migration artifacts. This paper presents a migration deconvolution method via deep learning based on the Hessian filter least-squares migration (HF-LSM) algorithm. The idea is to use deep learning techniques to model the inverse operator instead of directly estimating the inverse Hessian matrix. A data set is generated from a given velocity model by applying Born modeling to the migrated image, followed by application of the reverse time migration algorithm. The resultant data set is then used to train several neural network models. The networks learn the blurring operator that describes the image degradation due to the effects of acquisition geometry. Three different network topologies were developed to handle this problem: a simple fully convolutional neural network, a U-Net and a generative adversarial network. Our results show that the proposed approach provides images of higher resolution and superior quality than the traditional HF-LSM workflow.

The rigorous solution of the scattering problem for a finite cone embedded in a dielectric sphere surrounded by the dielectric medium

Wave scattering from a finite hollow cone with perfectly conducting boundaries embedded in a dielectric sphere is considered. The structure is excited axially symmetrically by the radial electric dipole. The scattering problem is formulated in the spherical coordinate system as the boundary value problem for the Helmholtz equation. The diffracted field is given by expansion in the series of eigenfunctions. Owing to the enforcement of the conditions of continuity together with the orthogonality properties of the Legendre functions the diffraction problem is reduced to infinite system of linear algebraic equations (ISLAE) of the first kind. The usage of the analytical regularization approach transforms the ISLAE of the first kind to the second one and allows one to justify the truncation method for obtaining the numerical solution in the required class of sequences. This system is proved to be regularized by a pair of operators, which consist of the convolution type operator and the corresponding inverse one. The inverse operator is found analytically using the factorization technique. The numerical examples are presented. The static and the low-frequency approximations as well as the transition to the limiting case when the cone degenerates into the disc are considered.

Fractional dark energy

In this paper we introduce the fractional dark energy model, in which the accelerated expansion of the Universe is driven by a nonrelativistic gas (composed by either fermions or bosons) with a noncanonical kinetic term. The kinetic energy is inversely proportional to the cube of the absolute value of the momentum for a fluid with an equation of state parameter equal to minus one, and whose corresponding energy density mimics the one of the cosmological constant. In the general case, the dark energy equation of state parameter (times three) is precisely the exponent of the momentum in the kinetic term. We show that this inverse momentum operator appears in fractional quantum mechanics and it is the inverse of the Riesz fractional derivative. The observed vacuum energy can be obtained through the integral of the Fermi-Dirac (or Bose-Einstein) distribution and the lowest allowed energy of the particles. Finally, a possible thermal production and fate of fractional dark energy is investigated.

Inverse characterization of UHPC material based on Hopkinson bar test

Coupling scientific researches with engineering applications leads to significant increase in development of innovative materials. Since time-consuming and expensive experimental tests are required for characterization of materials under dynamic loading regime, utilizing inverse characterizations of material has been increased. In this paper, an effective inverse methodology is proposed and it is utilized to determine dynamic behavior of cement-based composite material. This inverse procedure is based on the experimental test via split Hopkinson pressure bar which is one of the most widely used devices in investigation of material response to high rate loading. In the current study, Holmquist–Johnson–Cook constitutive model is used to present dynamic behavior of ultra-high performance concrete that is a new generation of concrete and has higher strength compared to traditional concrete. The reflected and transmitted waves obtained from the experiments are utilized as input in the inverse procedure. Moreover, genetic algorithm with intergeneration projection is adopted as an inverse operator to determine inversed parameters. Based on the obtained results, the recommended characterization method indicates effectiveness in the material characterization. The proposed methodology can accurately estimate the desired parameters of material under the high strain rates.

ФУНКЦИИ ГРИНА НЕКОТОРЫХ КРАЕВЫХ ЗАДАЧ ДЛЯ БИГАРМОНИЧЕСКИХ ОПЕРАТОРОВ И ИХ КОРРЕКТНЫЕ СУЖЕНИЯ

In this paper, a constructive method is given for constructing the Green function of the Dirichlet problem for a biharmonic equation in a multidimensional ball. The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise. There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the construction of the Green function is an open problem. For the ball, the Green function of the internal and external Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases. Finding general correct boundary value problems for differential equations is always an urgent problem. The abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of symmetric operators with finite defect indices was developed in detail. Many problems for partial differential equations lead to operators with infinite defect indices. In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain minimum operator, independently of each other, in terms of the inverse operator. In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's function.

Adaptive Inverse Compensation for Unknown Input and Output Hysteresis Using Output Feedback Neural Control

The search for new approaches for output feedback control of uncertain nonlinear systems with unknown input and output hysteresis is an interesting problem in control theory. One challenging issue obstructs the development of output feedback control design is that both the genuine system input and output are unknown signals and unable to be employed in the observer and controller design. To obviate such obstruction, a new control design framework for adaptively compensating the input and output hysteresis is proposed with two adaptive hysteresis inverse operators, which are also utilized to develop a novel adaptive hysteresis operator-based filter. It is proved that with the proposed control scheme, all the closed-loop signals are bounded and the tracking error ultimately converges to a tunable residual around zero. Simulation studies demonstrate the methods developed.

Appell and Sheffer sequences: on their characterizations through functionals and examples

The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several examples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite $d$-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials.

Analytical Inversion of the Operator Matrix for the Problem of Diffraction by a Cylindrical Segment in Sobolev Spaces

A vector problem of electromagnetic-wave diffraction by a cylinder is described by a system of two two-dimensional integro-differential equations. After expanding the unknown functions and the right-hand sides in Fourier series, the problem reduces to systems of one-dimensional equations. Analytical inversion of the principal operator of one-dimensional systems in Sobolev spaces is considered. Theorems on the boundedness and bounded invertibility of the principal operator are proved. The inverse operator is represented by series and in closed form: the elements of the inverse matrix are integral or integro-differential operators.

On the generalization of moyal equation for an arbitrary linear quantization

For an arbitrary linear combination of quantizations, the kernel of the inverse operator is constructed. An equation for the evolution of the Wigner function for an arbitrary linear quantization is derived and it is shown that only for Weyl quantization this equation does not contain a source of quasi-probability. Stationary solutions for the Wigner function of a harmonic oscillator are constructed, depending on the characteristic function of the quantization rule. In the general case of Hermitian linear quantization these solutions are real but not positive. We found the representation of Weyl quantization in the form of the limit of a sequence of linear Hermitian quantizations, such that for each element of this sequence the stationary solution of the Moyal equation is positive.